Intermediate mathematics
See also: Category:Foundations A generalization (or generalisation) is the formulation of general concepts from specific instances by abstracting common properties. Generalization is the process of identifying the parts of a whole, as belonging to the whole. The parts, completely unrelated may be brought together as a group by establishing a common relation between them.Wikipedia:Generalization Numbers The basis of all of mathematics is the "Next" function (see Graph theory). Next(0)=1, Next(1)=2, Next(2)=3, Next(3)=4...This defines the whole numbers. Repeatedly calling the Next function is defined as addition and its inverse is subtraction. But this leads to the ability to write equations like 1-3=x for which there is no answer among the whole numbers. To provide an answer mathematicians generalize the idea of whole numbers to the set of all integers which includes negative integers. Repeated addition is defined as multiplication and its inverse is division. But this leads to equations like 3/2=x for which there is no answer among integers. So mathematicians generalize the idea of integers to the idea of rational numbers. Repeated multiplication is defined as Exponentiation and its inverse is the nth root. But this leads to 2 problems: :Equations like sqrt(2)=x for which there is no answer among rational numbers so mathematicians generalize the idea of rational numbers to the idea of real numbers. :Equations like sqrt(-1)=x for which there is no answer among real numbers so mathematicians generalize the idea of real numbers to the idea of complex numbers. Complex numbers can be used to represent and actually perform rotations but only in 2 dimensions. Tensors, on the other hand, can be used to represent and perform rotations (and other linear transformations) in any number of dimensions. Rotations in n dimensions are called SO(n). See Graphical explanation of Tensor components. A tensor is a multivector. Understanding how tensors work leads to geometric algebra. Clifford algebra is a generalization of geometric algebra. Geometry :(In the discussion that follows {\mathbf e}_1 , {\mathbf e}_2 , {\mathbf e}_3 are orthogonal unit basis vectors) The one dimensional number line can be generalized to the idea of multidimensional math (i.e. geometry). Coordinate systems define the length of vectors parallel to one of the axes but leave all other lengths undefined. This concept of "length" which only works for certain vectors is generalized to the idea of the "norm" which works for all vectors. The norm of vector v is \|\mathbf{v}\| . In Euclidean space \|\mathbf{v}\|^2 = v_1^2 + v_2^2 + v_3^2 . See Pythagorean theorem. A topological space is a generalization of a metric space which is a generalization of a normed vector space. :A manifold is a generalization of Euclidean space. Multiplication can be generalized to allow for Multiplication of vectors in 4 different ways: Outer product (a tensor): \mathbf{u} \otimes \mathbf{v} . :As one would expect, every component of one vector multipies with every component of the other vector. : \begin{align}\mathbf{u} \otimes \mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 \mathbf{e_1} \mathbf{e_1} & u_1 v_2 \mathbf{e_1} \mathbf{e_2} & u_1 v_3 \mathbf{e_1} \mathbf{e_3} \\ u_2 v_1 \mathbf{e_2} \mathbf{e_1} & u_2 v_2 \mathbf{e_2} \mathbf{e_2} & u_2 v_3 \mathbf{e_2} \mathbf{e_3} \\ u_3 v_1 \mathbf{e_3} \mathbf{e_1} & u_3 v_2 \mathbf{e_3} \mathbf{e_2} & u_3 v_3 \mathbf{e_3} \mathbf{e_3} \end{bmatrix} \end{align} :And more simply: : \mathbf{u} \otimes \mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} \\ u_2 \mathbf{e_2} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} \\ u_3 \mathbf{e_3} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} \end{bmatrix} ::The Tensor product generalizes the outer product. The tensor product of a rank n tensor and a rank m tensor results in a rank n+m tensor. Dot product (a Scalar): \mathbf{u}\bullet\mathbf{v} = \|\mathbf{u}\|\ \|\mathbf{v}\|\cos(\theta) = u_1 v_1 + u_2 v_2 + u_3 v_3 : \mathbf{u}\bullet\mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 \mathbf{e_1}\mathbf{e_1} + u_2 v_2 \mathbf{e_2}\mathbf{e_2} + u_3 v_3 \mathbf{e_3}\mathbf{e_3} \end{bmatrix} :Strangely, only parallel components multiply. Or maybe not so strange, after all, if one of the vectors is actually a covector. :In Euclidean space \|\mathbf{v}\|^2 = \mathbf{v}\bullet\mathbf{v} . :The dot product can be generalized to the bilinear form (B(u,v) = scalar) and its associated quadratic form (Q(x) = B(x,x)). The dot product of a rank n tensor and a rank m tensor results in a rank n-m tensor. ::The bilinear form can be further generalized to the Sesquilinear form (an inner product is a sesquilinear form). Wedge product (a simple bivector): \mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} : :The wedge product is also called the exterior product (sometimes mistakenly called the outer product). The term "exterior" comes from the exterior product of two vectors not being a vector. In three dimensions \mathbf{u} \wedge \mathbf{v} is a pseudovector and its dual is \mathbf{u} \times \mathbf{v} (cross product). See also Regressive product. : \mathbf{u} \wedge \mathbf{u} = 0 Geometric product (a multivector): \mathbf{u} \mathbf{v} = \mathbf{u} \bullet \mathbf{v} + \mathbf{u} \and \mathbf{v} : :(The diagonals of the tensor above correspond to the dot product). : \mathbf{e_1} \mathbf{e_1} = \mathbf{e_{11}} = \mathbf{e_1} \bullet \mathbf{e_1} + \mathbf{e_1} \and \mathbf{e_1} = 1 + 0 = 1 : \mathbf{e_2} \mathbf{e_2} = \mathbf{e_{22}} = \mathbf{e_2} \bullet \mathbf{e_2} + \mathbf{e_2} \and \mathbf{e_2} = 1 + 0 = 1 : \mathbf{e_3} \mathbf{e_3} = \mathbf{e_{33}} = \mathbf{e_3} \bullet \mathbf{e_3} + \mathbf{e_3} \and \mathbf{e_3} = 1 + 0 = 1 Multiplication of a vector and a scalar belonging to a field can be generalized to multiplication of a vector and a scalar belonging to a ring. The result is a module (as opposed to a vector space). A mapping is a generalization of a function. A morphism (This leads to category theory) is a generalization of a homomorphism which is a generalization of a linear map which is a generalization of a linear transformation. Tensors seem to be generalizations of multilinear maps. Integration The integral (antiderivative) is a generalization of multiplication. :For example: an object dropped from point r1 to point r2 will release energy but the usual equation mass \cdot gravity \bullet (r_1 - r_2) = energy cant be used if the strength of gravity is itself a function of radius. The strength of gravity at r1 would be different than is is at r2. And in fact g® = 1/r^2 (See inverse-square law.) :However, the corresponding Definite integral is easily solved: mass \cdot \int_{r_1}^{r_2} g® \cdot dr : The derivative is a generalization of division. The derivative of the integral of f(x) is just f(x). Partial derivatives and multiple integrals generalize derivatives and integrals to multiple dimensions. The partial derivative with respect to one variable \frac{\part f(x,y)}{\part x} is found by simply treating all other variables as though they were constants. Multiple integrals are found the same way. : The Lie derivative generalizes the Lie bracket which is a generalization of the cross product. The cross product is neither commutative nor associative and therefore doesnt form a field or even a ring (see below). Instead it forms a Lie algebra which is a local or linearized version of a Lie group. A Lie group is a group that is also a differentiable manifold. Generalization of addition and multiplication :Main article: Algebraic structure Addition and multiplication can be generalized in so many ways that mathematicians were forced to create categories to organize them. Dot product maybe not so strange :See Hodge dual Let \mathbf\bar be a covector (a pseudovector) that is the orthogonal complement of vector \mathbf{v} . (A covector would be a tensor of rank -1.) \mathbf{u} \bullet \mathbf{v} = \mathbf{u} \otimes \mathbf\bar = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_2 e_3} & v_2 \mathbf{e_1 e_3} & v_3 \mathbf{e_1 e_2} \end{bmatrix} Therefore: \mathbf{u} \bullet \mathbf{v} = \begin{bmatrix}u_1 v_1 \mathbf{e_1 e_2 e_3} & u_1 v_2 \mathbf{e_1 e_1 e_3} & u_1 v_3 \mathbf{e_1 e_1 e_2} \\ u_2 v_1 \mathbf{e_2 e_2 e_3} & u_2 v_2 \mathbf{e_2 e_1 e_3} & u_2 v_3 \mathbf{e_2 e_1 e_2} \\ u_3 v_1 \mathbf{e_3 e_2 e_3} & u_3 v_2 \mathbf{e_3 e_1 e_3} & u_3 v_3 \mathbf{e_3 e_1 e_2} \end{bmatrix} Which reduces to: \mathbf{u} \bullet \mathbf{v} = \begin{bmatrix}u_1 v_1 & 0 & 0 \\ 0 & u_2 v_2 & 0 \\ 0 & 0 & u_3 v_3 \end{bmatrix} = u_1 v_1 + u_2 v_2 + u_3 v_3 Because: : \mathbf{e_1 e_2 e_3} = 1 = unit trivector which in 3 dimensions is a pseudoscalar. : \mathbf{e_1 e_2 e_2} = 0 = trivector with zero volume (since its 2 dimensional). References This article incorporates text from Wikipedia:Curl_(mathematics)#Generalizations Category:Foundations